Investigating the proof of lem:descent lemma in [1]}
\(J(u_1) & = J(u_2) + \int _0^1\left\langle \nabla J(u_2)+t(u_1-u_2)),u_1-u_2\right\rangle \mathrm {d}t \\&= J(u_2) + \left\langle \nabla J(u_2),u_1-u_2\right\rangle \\&\quad \quad +\int _0^1 \left\langle \nabla J\big (u_2+t(u_1-u_2)\big )-\nabla J(u_2),u_1-u_2\right\rangle \mathrm {d}t\\&\le J(u_2) + \left\langle \nabla J(u_2),u_1-u_2\right\rangle \\&\quad \quad +\int _0^1 \left\Vert \nabla J\big (u_2+t(u_1-u_2)\big )-\nabla J(u_2)\right\Vert \cdot \left\Vert u_1-u_2\right\Vert \mathrm {d}t \\&\le J(u_2) + \left\langle \nabla J(u_2),u_1-u_2\right\rangle + \int _0^1 Lt\Vert u_1-u_2\Vert ^2\mathrm {d}t \\&= J(u_2) + \left\langle \nabla J(u_2),u_1-u_2\right\rangle + \tfrac{L}{2}\Vert u_1-u_2\Vert ^2,\) 
